অনুশীলনী – ৫.৩ উৎপাদকে বিশ্লেষণ সপ্তম শ্রেণির গণিত সমাধান | PDF| School| Madrasah

\begin{align*}
&\mathbf{১। x²+xy+zx+yz}\\
&সমাধান :\\
&x²+xy+zx+yz\\
&=x\left(x+y\right)+z\left(x+y\right)           [কমন]\\
&=\left(x+y\right)\left(x+z\right)       [কমন]\\
&\mathbf{২। a²+bc+ca+ab}\\
&সমাধান :\\
&a²+bc+ca+ab\\
&=a\left(a+c\right)+b\left(c+a\right)        [কমন]\\
&=a\left(a+c\right)+b\left(a+c\right)\\
&=\left(a+b\right)\left(a+c\right)        [কমন]\\
&\mathbf{৩। ab\left(px+qy\right)+a²qx+b²py}\\
&সমাধান :\\
&ab\left(px+qy\right)+a²qx+b²py\\
&=abpx+abqy+a²qx+b²py\\
&=abpx+a^{2}qx+b^{2}py+abqy\\
&=ax\left(bp+aq\right)+by\left(bp+aq\right)                  [কমন]\\
&=\left(bp+aq\right)\left(ax+by\right)                    [কমন]\\
&\mathbf{৪। 4x²-y²}\\
&সমাধান :\\
&4x²-y²\\
&=\left(2x\right)²-y²\\
&=\left(2x+y\right)\left(2x-y\right)         \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\

&\mathbf{৫। 9a²-4b²}\\
&সমাধান :\\
&9a²-4b²\\
&=\left(3a\right)²-\left(2b\right)²\\
&=\left(3a+2b\right)\left(3a-2b\right)          \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&\mathbf{৬। a²b²-49y²}\\
&সমাধান :\\
&a²b²-49y²\\
&=\left(ab\right)²-\left(7y\right)²\\
&=\left(ab+7y\right)\left(ab-7y\right)             \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&\mathbf{৭। 16x⁴-81y⁴}\\
&সমাধান :\\
&16x⁴-81y⁴\\
&=\left(4x²\right)²-\left(9y²\right)²\\
&=\left(4x²+9y²\right)\left(4x²-9y²\right)               \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&=\{\left(2x\right)²-\left(3y\right)²\}\left(4x²+9y²\right)\\
&=\left(2x+3y\right)\left(2x-3y\right)\left(4x²+9y²\right)            \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&\mathbf{৮। a²-\left(x+y\right)²}\\
&সমাধান :\\
&a²-\left(x+y\right)²\\
&=a²-\left(x+y\right)²\\
&=\{a+\left(x+y\right)\}\{a-\left(x+y\right)\}         \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&=\left(a+x+y\right)\left(a-x-y\right)\\
&\mathbf{৯। \left(2x-3y+5z\right)²-\left(x-2y+3z\right)²}\\
&সমাধান :\\
&\left(2x-3y+5z\right)²-\left(x-2y+3z\right)²\\
&=\{2x-3y+5z+\left(x-2y+3z\right)\}\{2x-3y+5z-\left(x-2y+3z\right)\}   \\
&\left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&=\left(2x-3y+5z+x-2y+3z\right)\left(2x-3y+5z-x+2y-3z\right)\\
&=\left(3x-5y+8z\right)\left(x-y+2z\right)\\
&\mathbf{১০। 4+8a²+9a⁴}\\
&সমাধান :\\
&4+8a²+9a⁴\\
&=9a⁴+8a²+4\\
&=  9a^{4}+12a^{2}+4-4a^{2}\\
&= {\left(3a^{2}\right)}^{2}+2.3a.2+2^{2}-4a^{2}\\
&=  {\left(3a^{2}+2\right)}^{2}-{\left(2a\right)}^{2}           \left[সূত্র\ {\left(a+b\right)}^{2}=a^{2}+2ab+b^{2}\right]\\
&=  \left(3a^{2}+2+2a\right)\left(3a^{2}+2-2a\right)        \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&=  \left(3a^{2}+2a+2\right)\left(3a^{2}-2a+2\right)\\
&\mathbf{১১। 2a²+6a-80}\\
&সমাধান :\\
&2a²+6a-80\\
&=2\left(a²+3a-40\right)                    [কমন]\\
&=2\left(a²+8a-5a-40\right)           [ মিডল টার্ম ব্রেক ]\\
&=2\{a\left(a+8\right)-5\left(a+8\right)\}                    [কমন]\\
&=2\left(a+8\right)\left(a-5\right)                    [কমন]\\
&\mathbf{১২। y²-6y-91}\\
&সমাধান :\\
&y²-6y-91\\
&=y²-13y+7y-91       [ মিডল টার্ম ব্রেক ]\\
&=y\left(y-13\right)+7\left(y-13\right)                        [কমন]\\
&=\left(y-13\right)\left(y+7\right)                    [কমন]\\
&\mathbf{১৩। p²-15p+56}\\
&সমাধান :\\
&p²-15p+56\\
&=p²-8p-7p+56         [ মিডল টার্ম ব্রেক ]\\
&=p\left(p-8\right)-7\left(p-8\right)                        [কমন]\\
&=\left(p-8\right)\left(p-7\right)                    [কমন]\\
&\mathbf{১৪। 45a⁸-5a⁴x⁴}\\
&সমাধান :\\
&45a⁸-5a⁴x⁴\\
&=5a⁴\left(9a⁴-x⁴\right)             [কমন]\\
&=5a⁴\{\left(3a²\right)²-\left(x²\right)²\}\\
&=5a⁴\left(3a²+x²\right)\left(3a²-x²\right)\ \ \ \ \ \ \ \ \ \ \ \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&\mathbf{১৫। a²+3a-40}\\
&সমাধান :\\
&a²+3a-40\\
&=a²+8a-5a-40           [ মিডল টার্ম ব্রেক ]\\
&=a\left(a+8\right)-5\left(a+8\right)              [কমন]\\
&=\left(a+8\right)\left(a-5\right)                [কমন]\\
&\mathbf{১৬। \left(x²+1\right)²-\left(y+1\right)²}\\
&সমাধান :\\
&\left(x²+1\right)²-\left(y+1\right)²\\
&=\{x²+1+\left(y+1\right)\}\{x²+1-\left(y+1\right)\}   \ \ \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&=\left(x^{2}+1+y+1\right)\left(x^{2}+1-y-1\right)\\
&=\left(x²+y+2\right)\left(x²-y\right)\\

&\mathbf{১৭। x²+11x+30}\\
&সমাধান :\\
&x²+11x+30\\
&=x²+6x+5x+30          [ মিডল টার্ম ব্রেক ]\\
&=x\left(x+6\right)+5\left(x+6\right)              [কমন]\\
&=\left(x+6\right)\left(x+5\right)              [কমন]\\
&\mathbf{১৮। a²-b²+2bc-c²}\\
&সমাধান :\\
&a²-b²+2bc-c²\\
&=a²-\left(b²-2bc+c²\right)\\
&=a²-\left(b-c\right)²          [সূত্র\ {\left(a-b\right)}^{2}=a^{2}-2ab+b^{2}]\\
&=\{a+\left(b-c\right)\} \{a-\left(b-c\right)\}       \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&=\left(a+b-c\right)\left(a-b+c\right)\\
&\mathbf{১৯। 144x⁷-25x³a⁴}\\
&সমাধান :\\
&144x⁷-25x³a⁴\\
&=x³\left(144x⁴-25a⁴\right)             [কমন]\\
&=x³\{\left(12x²\right)²-\left(5a²\right)²\}\\
&=x³\left(12x²+5a²\right)\left(12x²-5a²\right)             \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&\mathbf{২০। 4x²+12xy+9y²-16a²}\\
&সমাধান :\\
&4x²+12xy+9y²-16a²\\
&=\left(2x\right)²+2.\left(2x\right).\left(3y\right)+\left(3y\right)²-16a²      \\ 
&=\left(2x+3y\right)²-\left(4a\right)²            \left[সূত্র\ {\left(a+b\right)}^{2}=a^{2}+2ab+b^{2}\right]\\
&=\left(2x+3y+4a\right)\left(2x+3y-4a\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]
\end{align*}